Modern Algebra, Group Theory (XXI): Abelian Group:Prove that a group G is abelian if every element , except the identity, is of order 2.

Modern Algebra Group Theory (XXI) Abelian Group Prove that a group G is abelian if every element , except the identity, is of order 2.

Modern Algebra, Group Theory (XXII): Group of Even Order:If G is a group of even order, prove it has an element a ≠ e satisfying a^2 = e.

Modern Algebra Group Theory (XXII) Group of Even Order If G is a group of even order, prove it has an element a ≠ e satisfying a^2 = e.

Modern Algebra, Group Theory (XXIII): Formation of a Group.

Modern Algebra Group Theory (XXIII) Formation of a Group Let G be a nonempty set closed under an associative product, which in addition satisfies: (a) There exists an ...continues

Modern Algebra, Group Theory (XXIV): Formation of a Group.

Modern Algebra Group Theory (XXIV) Formation of a Group Let G be a nonempty set closed under an associative product, which in addition satisfies: (a) Th ...continues

Modern Algebra, Group Theory (XXV): Formation of a Group.

Modern Algebra Group Theory (XXV) Formation of a Group Let G be a nonempty set closed under an associative product, which in addition satisfies: (a) Ther ...continues

Modern Algebra, Group Theory (XXVI): Subgroups of a Group: If H and K are subgroups of G, then so is H∩K a subgroup of G.

Modern Algebra Group Theory (XXVI) Subgroups of a Group If H and K are subgroups of G, then so is H∩K a subgroup of G.

Question about group theory, cardinality and isomorphic

I would like to know how to identify and prove the cardinality of sets and how to identify isomorphic. (See attached file for full problem description) --- Group Theory: a. If S and T are sets then let TS denote the set of all functions from S to T. Prove that the cardinality of TSxU equals the cardinality of (TS)U ...continues

Group Structure, Order of two groups

I have questions about constructing a group structure, how to identify the order of a paired group when they have different orders and method of figuring out the group identity and the inverse of a pair that contained in the paired group. --- If G and H are groups then explain how to equip G x H with a group structure. If G ...continues

For a subgroup H of G define a left coset of H in G as the set of all elements of the form ah, hєH.Show that there is a 1-1 correspondence between the set of left cosets of H in G and the set of right cosets of H in G.

Modern Algebra Group Theory (XXVII) Subgroups of a Group Cosets of Subgroups of a Group For a subgroup H of G define a left coset of H in G as the set of all ...continues

For a subgroup H of G define a left coset of H in G as the set of all elements of the form ah, hєH.Show that there is a 1-1 correspondence between the set of left cosets of H in G and the set of right cosets of H in G.

Modern Algebra Group Theory (XXVII) Subgroups of a Group Cosets of Subgroups of a Group For a subgroup H of G define a left coset of H in G as the set of all ...continues